Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 371342, 8 pages http://dx.doi.org/10.1155/2015/371342 Research Article A Novel Analytical Solution Method for Constraint Forces of the Kinematic Pair and Its Applications Changjian Zhi, Sanmin Wang, Yuantao Sun, and Bo Li School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China Correspondence should be addressed to Changjian Zhi; [email protected] Received 23 December 2014; Revised 11 March 2015; Accepted 12 March 2015 Academic Editor: Vladimir Turetsky Copyright © 2015 Changjian Zhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Constraint forces of the kinematic pair are the basis of the kinematics and dynamics analysis of mechanisms. Exploring the solution method for constraint forces is a hot issue in the mechanism theory fields. Based on the observation method and the theory of reciprocal screw system, the solution method of reciprocal screw system is improved and its solution procedures become easier. This method is also applied to the solution procedure of the constraint force. The specific expressions of the constraint forceare represented by the reciprocal screw system of twist. The transformation formula of twist under different coordinates is given and it make the expression of the twist of kinematic pair more facility. A slider-crank mechanism and a single loop spatial RUSR mechanism are taken as examples. It confirms that this method can be used to solve the constraint force of the planar and spatial mechanism. 1. Introduction deformation compatibility method. Russo et al. [7]using the counterweight method and the springs method analyzed The constraint force analysis of kinematic pair is not only the static balancing of spatial parallel manipulator. Lu [8] thekeyofusingmechanismsreasonablyandcreatingnew used virtual work theory and CAD functionalities for solving mechanisms but also the important factor of kinematics active force and passive force of spatial parallel manipulators. and dynamics analysis and the base of structure design of Deepak and Ananthasuresh [9]usednonlinearspringsto mechanism. Traditional methods of constraint force analysis generate minimum torques which keep the cables taut and of kinematic pair include the graphic method and analytical analyzed the static balancing of the parallel cable-driven method [1]. Since they need drawing and the force figures, mechanisms. which go against programming, seeking new analysis meth- The reciprocal screw system represents the constraints ods is very necessary. Many scholars have a try in this filed and have got some achievements. Zhou et al. [2]utilizingthe and constraint forces acting on kinematic pairs. Since its dismantle-bar method and the microdeformation and super- solution method is complex and lacks commonality, based position principle analyzed the static force of the parallel on Huang et al. [10] and Dai and Jones [11, 12]researches, mechanism. Zhao and Huang [3]usingtheanalyticalmethod we improve the solution method of the reciprocal screw analyzed the force of the lower-mobility parallel mechanism system and make the solving process easier. In order to with overconstrained couple. Based on Zhao’s research and express constraint forces of the kinematic pair, the solu- screw theory, Liu et al. [4]analyzedtheforceofsingleloop tion coefficient is introduced. Not only can this method spatial mechanism. Wang et al. [5] combining the traditional solve constraint forces of the kinematic pair of mechanism dismantle-bar method with screw theory analyzed the force but also it can analyze the static and dynamic force by of the spatial parallel mechanism. Jiang et al. [6]using combining it with the foregoing methods. We verify the Newton Euler method and D’Alembert principle established method by solving constraint forces of the kinematic pair the force analysis equations, and they also put forward the of a slider-crank mechanism and a single loop spatial RUSR dynamic analysis model of parallel mechanism based on the mechanism. 2 Mathematical Problems in Engineering z S0 S S hS r S0 −hS x y Figure 2: The axis of a screw. Figure 1: A line vector. where is the instantaneous angular velocity of the rigid 0 2. Theoretic Foundation body and k is the instantaneous velocity a point in the rigid body coincident with the origin. Alinevector(Figure 1)canbedenotedbyadualvector[10] According to formula (4), S canbewrittenas S =(S; S )=(S; r × S) , (1) 0 0 S =(; k −ℎ)+(0;ℎ) , (6) (S; S ) S where 0 is called the Plucker¨ coordinates, is the real 0 unit, which is the direction ratios of the line and is not origin- where v −ℎ = r × ; ℎ canbeobtainedbyformula(3). (; v0 −ℎ) dependent, S0 is the real unit, which is the moment of the line Therefore, canbeseenasapurerotation, about the origin and is origin-dependent, and r is the vector and (0;ℎ) can be seen as a pure translation. A twist can be of a point on the line. S and S0 are three-dimensional vectors. decomposed into a rotation and a translation. S and S0 satisfy the orthogonal condition, S⋅S0 =0.When Generally, all the spatial forces acting on a rigid body can 0 S =0, the line passes through the origin. While S =0,theline be reduced to a force (F; F0) and a couple (0; T ).Theforce lies in a plane at infinity, and it becomes a couple. and the couple may have different directions. According to If S and S0 do not satisfy the orthogonal condition, S⋅S0 ≠ thescrewalgebra,thesumoftheforceandthecoupleisa 0. S is known as a screw. Ball [13] described that a screw is a new screw, which is called a wrench. Consider straight line with the pitch. It can be expressed as 0 0 S =(F; F0 + T )=(F; C ). (7) 0 S =(S; S ), (2) According to formula (4), S canbewrittenas where S is the real unit, which is the direction ratios of 0 the screw axis and is not origin-dependent, S is the dual S =(F; C0 −ℎF)+(0;ℎF) , 0 (8) unit, which is not origin-dependent. S and S are three- 0 dimensional vectors. where C −ℎF = r × F; ℎ canbeobtainedbyformula(3). S =0̸ ℎ 0 If ,thepitchofascrewis : Therefore, (F; C −ℎF) canbeseenasaforce,and(0;ℎF) S ⋅ S0 canbeseenasacouple.Theyhavethesamedirection. ℎ= . (3) S ⋅ S 0 3. Expressions of Constraint Forces of the In order to decide the position of the axis of a screw, S Kinematic Pair of Mechanism by Screw canbedecomposedintotwoparts(Figure 2). One is parallel to S and the other is perpendicular to S. Consider The specific Plucker¨ coordinate of twist and wrench can be 0 written as [10] S =(S; S −ℎS)+(0;ℎS) . (4) S =(,,;∗,∗,∗), 0 0 (9) Since S −ℎS is perpendicular to S, S −ℎS = S0. 0 0 ∗ ∗ ∗ (S; S −ℎS) represents a line vector. (S;ℎS) represents a couple. where S =(,,), S =( , , ). Obviously, a screw can be decomposed into a line vector and If S represents a twist, (, ,)isitsangularvelocity, ∗ ∗ ∗ a couple. According to this formula, a line vector is a special and ( , , ) is its linear velocity. screw. If S represents a wrench, (, ,) is its force part, and ∗ ∗ ∗ The screw can be used to describe motions and forces. ( , , )isitscouplepart. They are, respectively, called twist and wrench. The instan- In order to calculate the reciprocal screw, a screw can be taneous twist of a rigid body can be written as expressed by a row vector: 0 ∗ ∗ ∗ S =(; k ), (5) S =[ ]. (10) Mathematical Problems in Engineering 3 A kinematic pair is a combination of two kinematic where G is a (6−)by(+1)matrix;C is a new bodies which have relative motion with respect to each other. matrix which is formed by deleting the th column of Its motion can be described by screw system. The order of the C matrix. The element of V in the th row and th screw system is the same as the mobility. If the mobility of a column is V. It satisfies the following relationship: kinematic pair is (1≤≤5, is an integer), its twist is expressed as V = G , (16) ∗ ∗ ∗ 1 1 1 =O [ 1 1 1 ] where . [ . ] S = [ . ] . The rest of elements of the th column of V equal 0. [ . ] (11) C ∗ ∗ ∗ If there is a zero row vector in the matrix ,itis [ ] removed forming a new matrix M. columns are M −1 N S Sr chose from which forms a ( )by matrix .The is a screw system, and its reciprocal screw system is . numerical order of the chosen columns in the matrix They satisfy the following relationship: S are saved in the th row of the matrix H which is a r (6−)by matrix. Consider S ∘ S =0, (12) ++ G = (−1) N (1≤≤), where ∘ is the reciprocal product. (17) r Let V = S Δ,whereΔ is transformation operator between the real unit vector and the dual unit vector. The where G is a (6−)by(+1)matrix;N is a new expression of Δ is matrix which is formed by deleting the th column of the N matrix. The element of V in the th row and th 0I column is V. It satisfies the following relationship: Δ=[ ], (13) I0 V = G, (18) where I is a 3 by 3 unit matrix. =H S and V satisfy the following relationship: where . V The rest of elements of the th column of equal 0. S V =0. (14) (iii) If V can satisfy the requirements, the solution pro- S V canberegardedasthenullspaceofthescrewsystems cedures of the reciprocal screw system of are V S . Based on Huang’s observation method and Dai’s linear finished. If cannot satisfy the requirements, the dissatisfactory row is removed and step (2) is repeated algebraic method obtaining reciprocal screw systems, we V improve the solution method of V.